Under acceleration and breaking my seats (both sides) move forwards and backwards. The movement is not great but it is noticeable. I have spoken to the previous owner and he mentioned it is the washers that have worn out and it's a cheap and easy job to repair.

Has anyone else experienced this problem, know how to fix it and where to source the parts required?

Search YouTube for bmw z3 seat bushes and there is a guy showing you how to do it.
While you have the seat out of the car don't turn the ignition on other wise you will have to have the light reset on the dash.

10 minutes to get everything ready.
5-10 to get the seat out of the car and on the bench.
Another 5-10 minutes to strip the seat.
10-15 to accurately sand down the spacers.
5-10 to re-assemble the seat.
5-10 to refit the seat.
10 minutes to clear up the mess and put the tools away.
10 minutes to make a cuppa.
3 hours for a test drive.

10 minutes to get everything ready.
5-10 to get the seat out of the car and on the bench.
Another 5-10 minutes to strip the seat.
10-15 to accurately sand down the spacers.
5-10 to re-assemble the seat.
5-10 to refit the seat.
10 minutes to clear up the mess and put the tools away.
10 minutes to make a cuppa.
3 hours for a test drive.

You should make the tea before you start
and then another one afterwards